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Function transformations refer to changes that affect the position and shape of a function's graph. These transformations can include shifts, stretches, or reflections. In our exercise, we observe vertical and horizontal shifts as the primary transformations.

When we take a function such as the quadratic function, like our given function \(g(x) = x^2\), a transformation occurs when another operation or function modifies it. For instance, the output \(f(g(x)) = x^2 - 7\) is a vertical transformation. Here, each point on the parabola shifts down by 7 units.

Similarly, in \(g(f(x)) = (x - 7)^2\), we have a horizontal shift. The entire graph of the quadratic function moves 7 units to the right. Recognizing these transformations is essential for understanding how composite functions behave.

A linear function, such as \(f(x) = x - 7\), is a foundational concept in mathematics. Linear functions are represented graphically by straight lines.

Key characteristics of a linear function include:

• Constant slope: This is the rate of change, and for \(f(x) = x - 7\), the slope is 1.
• Y-intercept: This is the point where the line crosses the y-axis. The y-intercept for \(f(x)\) is -7.

With each increase of 1 in \(x\), \(f(x)\) increases by 1, showcasing its uniform behavior. Hence, it results in a straight line that slants upward or downward depending on your point of view.

Quadratic functions, like \(g(x) = x^2\), are graphs shaped like a parabola. They can open upwards or downwards based on their coefficient. In this case, the parabola opens upwards because the coefficient of \(x^2\) is positive: 1.

Some features of quadratic functions are:

• Vertex: The turning point of the function. The vertex for \(g(x) = x^2\) is at the origin (0,0).
• Axis of symmetry: A line that divides the parabola into two mirror images, which for our \(g(x)\) case is the y-axis, \(x = 0\).

This structure of quadratic functions helps in recognizing transformations when implementing operations in composite functions.

Graphing functions is a visual representation of mathematical expressions, assisting in understanding behaviors and relationships between them.

When graphing \(f(x) = x - 7\), \(g(x) = x^2\), and their composites \(f \circ g(x)\) and \(g \circ f(x)\), observe:

• Straight lines for linear functions, indicating consistent change.
• Parabolas for quadratic functions, demonstrating rate changes.
• Shifts in the graph positions due to transformations.

Each visual can communicate attributes of functions clearly. For example, seeing that \(f \circ g(x) = x^2 - 7\) is a shifted parabola, and \(g \circ f(x) = (x-7)^2\) lies to the right of the origin, helps us to interpret the mathematical operations behind them.